Test 1 ECON3161, Game Theory Tuesday, September 2 th Directions: Answer each question completely. If you cannot determine the answer, explaining how you would arrive at the answer may earn you some points. 1. (30 points) Consider a simultaneous game between two rms where each can choose one of three di erent quantity levels: 30, 0, and 70. Their payo s are in the following matrix: q 2 = 30 q 2 = 0 q 2 = 70 q 1 = 30 1800; 1800 100; 2000 900; 1800 Firm 1 q 1 = 0 2000; 100 1600; 1600 800; 1200 q 1 = 70 1800; 900 1200; 800 0; 0 a ( points) De ne the term strictly dominant strategy. strategy? If so which rm and which strategy? Does either rm have a strictly dominant A strictly dominant strategy is a strategy that always provides a strictly greater payo than any other strategy the player could choose given the strategy the other player is choosing. In this game, 2000 is both Firm 1 s and s highest payo, so if any strategy is going to be strictly dominant for Firm 1 it has to be q 1 = 0 and for it would have to be q 2 = 0. We can see that neither strategy is strictly dominant because while q 1 = 0 and q 2 = 0 is best when the other rm chooses either 30 or 0, when the other rm chooses q i = 70 then choosing q j = 30 is best. b ( points) Looking at the entire 3x3 game (and just the entire game), does either rm have any strictly dominated strategies? If so, which rm and which strategy (or strategies) are strictly dominated and which strategy (or strategies) are they strictly dominated by? Yes, both rms have a strictly dominated strategy. For Firm 1, q 1 = 70 is strictly dominated by q 1 = 0 (2000>1800, 1600>1200, 800>0). For, q 2 = 70 is strictly dominated by q 2 = 0 (2000>1800, 1600>1200, 800>0) c (10 points) Use the iterated elimination of dominated strategies (IEDS) to reduce the matrix as far as possible. Explain the steps you use to reduce this matrix. From part b we know that we can eliminate q 1 = 70 because it is strictly dominated by q 1 = 0 as well as q 2 = 70 because it is strictly dominated by q 2 = 0. This leaves a 2x2 matrix: q 2 = 30 q 2 = 0 q 1 = 30 1800; 1800 100; 2000 Firm 1 q 1 = 0 2000; 100 1600; 1600 With this 2x2 matrix we can see that now q 1 = 30 is strictly dominated by q 1 = 0 and also q 2 = 30 is strictly dominated by q 2 = 0. Thus we can remove q 1 = 30 and q 2 = 30 to yield a 1x1 "matrix": 1
q 2 = 0 Firm 1 q 1 = 0 1600; 1600 This is the furthest the matrix can be reduced. d ( points) Find all pure strategy Nash equilibria (PSNE) to this game. Since the matrix can be reduced to a single outcome cell there is only one PSNE to this game: Firm 1 choose q 1 = 0 and choose q 2 = 0. e ( points) Will there be a mixed strategy Nash equilibrium to this game? If so nd it; if not, explain why not. There is no MSNE in this game. By using IEDS we can reduce the matrix to a single outcome cell so if we attempt to nd an MSNE to the game (either the 3x3 or 2x2 after q 1 = 70 and q 2 = 70 have been eliminated) we will be attempting to make players indi erent over strategies that are strictly dominated, which cannot happen. 2. (2 points) Consider the following game: P2 Y Z P1 W 2; 2 3 ; 3 X ; 1; 2 a (10 points) Find all pure strategy Nash equilibria (PSNE) to this game. The PSNE can be determined by using the method of best responses as in the game above. There are two PSNE: (1) P1 choose X, P2 choose Y and (2) P1 choose W and P2 choose Z. b (10 points) Find the mixed strategy Nash equilibrium (MSNE) to this game. The MSNE can be found be setting the expected value of each player s pure strategies equal to each other and then nding the probabilities of the other player that cause the expected value to be equal. For Pl s probabilities, letting w be the probability he chooses strategy W, we need: E 2 [Y ] = E 2 [Z] 2w + (1 w) = 3w + 2 (1 w) 3 3w = w 3 = w 3 = w So P1 would choose W with probability 3 and X with probability 1. of choosing strategy Y or Z when P1 uses these probabilities is 11. For P2 s probabilities, letting y be the probability he chooses strategy Y, we need: E 1 [W ] = E 1 [X] 2y + 3 (1 y) = y + 1 (1 y) 2 2y = 3y 2 = y 2 = y 2 Note that P2 s expected value
So P2 would choose Y with probability 2 and Z with probability 3. choosing strategy W or X when P2 uses these probabilities is 13. Note that P1 s expected value of So the MSNE is P1 choose W with probability 3 and X with probability 1 and P2 choose Y with probability 2 and Z with probability 3. c ( points) Can any of the equilibria (either pure or mixed) be eliminated using either the equilibrium payo dominance criterion or the undominated Nash equilibrium criterion? If so, which ones and by which criteria? None of the equilibria involve using a weakly dominated strategy so no equilibria can be removed by the undominated Nash equilibrium criterion. The table below lists the payo s for each player to each equilibrium: Eq. P1 payo P2 payo X; Y W; Z 3 3 MSNE 13 11 Since both players are strictly better o under the PSNE of X, Y, we can eliminate the other PSNE (W, Z) and the MSNE by equilibrium payo dominance. 3. (2 points) Consider a simultaneous game between two rms where each can choose one of three di erent price levels: $10, $0, and $70. Their payo s are in the following matrix: p 2 = $10 p 2 = $0 p 2 = $70 p 1 = $10 0 ; 0 0; 0 0; 0 Firm 1 p 1 = $0 0 ; 0 1600 ; 1600 3200 ; 0 p 1 = $70 0 ; 0 0; 3200 1800; 1800 a (10 points) Does any rm have any weakly dominated strategies? If so, which rm and which strategy (or strategies) are weakly dominated and which strategy (or strategies) are they weakly dominated by? Both rms have weakly dominated strategies in this game. For Firm 1, p 1 = $10 is weakly dominated by both p 1 = $0 and p 1 = $70. Also, p 1 = $70 is weakly dominated by p 1 = $0. The same is true for : p 2 = $10 is weakly dominated by both p 2 = $0 and p 2 = $70, and p 2 = $70 is weakly dominated by p 2 = $0. b (10 points) Find all pure strategy Nash equilibria (PSNE) to this game. The PSNE can be found by using the method of nding the best responses, as shown in the matrix above. There are two PSNE: (1) Firm 1 choosing p 1 = $10 and choosing p 2 = $10 and (2) Firm 1 choosing p 1 = $0 and choosing p 2 = $0. c ( points) Considering only the PSNE, can any equilibrium be eliminated using either the undominated Nash equilibrium or equilibrium payo dominance criteria? If so, which equilibrium and by which criterion? Explain why. The PSNE where both rms choose a price of $10 can be eliminated by either of the criteria. Note that choosing $10 is weakly dominated by choosing $0, so by undominated Nash equilibrium rms should choose the equilibrium where both choose a price of $0. Also, both rms receive a strictly higher payo in the PSNE when both choose $0, so by equilibrium payo dominance we can eliminate the equilibrium where both choose $10. 3
. (20 points) Consider the following 3 person game: Player 2 Player 2 A B A B Player 1 A 2 ; 1 ; 2 0; 1 ; 0 Player 1 A 0; 0; 1 0; 1 ; 1 B 1; 0; 0 1 ; 1 ; 0 B 1 ; 0; 1 1 ; 1 ; 1 - A B % Player 3 a ( points) Does any player have a strictly or weakly dominant strategy? which strategy? If so which player and The best responses are marked in the matrix above. Note that every payo for strategy B for Player 2 (in both matrices) is marked. Strategy B for Player 2 does strictly better than strategy A for Player 2 when: (1) Player 1 chooses B and Player 3 chooses A, (2) Player 1 chooses A and Player 3 chooses B, and (3) both Players 1 and 3 choose B. When both Players 1 and 3 choose A then strategy A and B (for Player 2) give the same payo of 1. So B is a weakly dominant strategy for Player 2. There are no other weakly dominant strategies. b (10 points) Find all pure strategy Nash equilibria (PSNE) to this game. There are two PSNE to this game: (1) All 3 players choose B and (2) all 3 players choose A. c ( points) Considering only the PSNE, can any equilibrium be eliminated using either the undominated Nash equilibrium or equilibrium payo dominance criteria? If so, which equilibrium and by which criterion? Explain why. The equilibrium when all 3 players choose B can be eliminated by equilibrium payo dominance because Players 1 and 3 are strictly better o when all choose A and Player 2 is no worse o when all choose A than when all choose B. Also note that the equilibrium when all 3 players choose A can be eliminated by undominated Nash equilibrium because Player 2 is using a weakly dominated strategy in this equilibrium. Bonus: ( points) Consider the game in question 3. Will there be any mixed strategy Nash equilibria in this game? Explain why or why not and nd the MSNE if you believe one exists. I m going to repost the matrix so it is here: p 2 = $10 p 2 = $0 p 2 = $70 p 1 = $10 0 ; 0 0; 0 0; 0 Firm 1 p 1 = $0 0 ; 0 1600 ; 1600 3200 ; 0 p 1 = $70 0 ; 0 0; 3200 1800; 1800 This is an odd game where there is a weakly dominant strategy for both rms where they each choose $0 and there are no strictly dominated strategies for any rm. Suppose we try to nd an MSNE over all 3 outcomes. Let a be the probability that Firm 1 chooses $10, b be the probability it chooses $0, and c be the probability it chooses $70. What I am going to do (you ll see why shortly) is set this up letting a = 1 b c. If I use the following two equations: and
I get: 0 (1 b c) + 0 b + 0 c = 0 (1 b c) + 1600 b + 3200 c 0 = 1600b + 3200c 3200c = 1600b 2c = b (I used a = 1 b c because I knew a would be multiplied by 0). Now there s a problem already, that 2c = b and the only way this can happen is if b = c = 0. Let s look at the second equation: 0 (1 b c) + 0 b + 0 c = 0 (1 b c) + 0 b + 1800 c 0 = 1800c 0 = c We have now con rmed that c = 0, which means that b = 0, which means that a = 1. But this leads us right back to our PSNE where both players choose $10. What if we wanted to try to make a mixture over only two strategies (say $10 and $0, or $10 and $70, or $0 and $70). If we try to mix over only $0 or $70 (and leave out the $10 strategy) then playing $70 is strictly dominated by $0 and we can t make a mixture that adheres to the laws of probability. If we try to mix over $10 and $0 (which makes the most sense since this involves the two strategies that determine the PSNE), then we get: 0 = 0 (1 b) + 1600 b 0 = b So we are right back to where we started with both rms choose a price of $10. If we try to mix over $10 and $70 (which may make the least sense since we are omitting the weakly dominant strategy), we still end up back at the PSNE where both players choose $10. 0 = 0 (1 c) + 1600 c 0 = c Now we have exhausted all the possibilities and we still cannot nd an MSNE that is not a PSNE. This is because there is a PSNE involving both rms using weakly dominated strategies. In class I mentioned that there are typically an odd number of equilibria in games - however, this is one of the special cases where we actually have an even number of equilibria as there are no MSNE which are not PSNE.